Please may you kindly assist me on this integration exercise: Suppose for some fixed real $a, b$ with $a \neq 0$, we have:

$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\log \sqrt x)x^{-b} \mathrm{d}x $$

Where $f(x) = \sum_{n=0}^\infty n^{2}\pi (n^{2}\pi - \frac{3}{2x})e^{-n^{2}\pi x + \frac{3}{2}\ln x}$.

Can this equality be possible for any nonzero $b$  ?

Please may you kindly assist me on this integration exercise: For real $a, b$ with $a \neq 0$, consider the equality $$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\log \sqrt x)x^{-b} \mathrm{d}x, $$ where $f(x) := \sum_{n=0}^\infty n^{2}\pi (n^{2}\pi - \frac{3}{2x})e^{-n^{2}\pi x + \frac{3}{2}\ln x}$.

Can this equality hold for a nonzero $b$?

Please may you kindly assist me on this integration exercise: Suppose for some fixed real $a, b$ with $a \neq 0$, we have:

$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\log \sqrt x)x^{-b} \mathrm{d}x $$

Where $f(x) = \sum_{n=0}^\infty n^{2}\pi (n^{2}\pi - \frac{3}{2x})e^{-n^{2}\pi x + \frac{3}{2x}}$.

Can this equality be possible for any nonzero $b$ ?

Please may you kindly assist me on this integration exercise: Suppose for some fixed real $a, b$ with $a \neq 0$, we have:

$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\log \sqrt x)x^{-b} \mathrm{d}x $$

Where $f(x) = \sum_{n=0}^\infty n^{2}\pi (n^{2}\pi - \frac{3}{2x})e^{-n^{2}\pi x + \frac{3}{2}\ln x}$.

Can this equality be possible for any nonzero $b$ ?

Please may you kindly assist me on this integration exercise: Suppose for some fixed real $a, b$ with $a \neq 0$, we have:

$$\int_1^N f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^N f(x)\sin(a\log \sqrt x)x^{-b} \mathrm{d}x $$

Where $f(x) = \sum_{n=0}^\infty n^{2}\pi (n^{2}\pi -3/2x)e^{-n^{2}\pi x + 3/2x}$.

Can this equality be possible for any nonzero $b$ ?

Please may you kindly assist me on this integration exercise: Suppose for some fixed real $a, b$ with $a \neq 0$, we have:

$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\log \sqrt x)x^{-b} \mathrm{d}x $$

Where $f(x) = \sum_{n=0}^\infty n^{2}\pi (n^{2}\pi - \frac{3}{2x})e^{-n^{2}\pi x + \frac{3}{2x}}$.

Can this equality be possible for any nonzero $b$ ?